Local Density Functional Theory for Superfluid Fermionic Systems

1. Local Density Functional Theory for Superfluid Fermionic Systems • The Unitary Fermi Gas • Bulgac, University of Washington • arXiv:cond-mat/0703526, • PRA, R , in press (2007)

2. Unitary Fermi gas in a harmonic trap Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007)

3. Outline: • What is a unitary Fermi gas • Very brief/skewed summary of DFT • Bogoliubov-de Gennes equations, renormalization • Superfluid Local Density Approximation (SLDA) • for a unitary Fermi gas • Fermions at unitarity in a harmonic trap within SLDA and comparison with ab intio results

4. What is a unitary Fermi gas

5. Bertsch Many-Body X challenge, Seattle, 1999 What are the ground state properties of the many-body system composed of spin ½ fermions interacting via a zero-range, infinite scattering-length contact interaction. • In 1999 it was not yet clear, either theoretically or experimentally, • whether such fermion matter is stable or not. - systems of bosons are unstable (Efimov effect) - systems of three or more fermion species are unstable (Efimov effect) • Baker (winner of the MBX challenge) concluded that the system is stable. • See also Heiselberg (entry to the same competition) • Chang et al (2003) Fixed-Node Green Function Monte Carlo • and Astrakharchik et al. (2004) FN-DMC provided best the theoretical • estimates for the ground state energy of such systems. • Thomas’ Duke group (2002) demonstrated experimentally that such systems • are (meta)stable.

6. Consider Bertsch’s MBX challenge (1999): “Find the ground state of infinite homogeneous neutron matter interacting with an infinite scattering length.” • Carlson, Morales, Pandharipande and Ravenhall, PRC 68, 025802 (2003), with Green Function Monte Carlo (GFMC) normal state • Carlson, Chang, Pandharipande and Schmidt, PRL 91, 050401 (2003), with GFMC superfluid state This state is half the way from BCS→BEC crossover, the pairing correlations are in the strong coupling limit and HFB invalid again.

7. BCS side BEC side Solid line with open circles – Chang et al. PRA, 70, 043602 (2004) Dashed line with squares - Astrakharchik et al. PRL 93, 200404 (2004)

8. Green Function Monte Carlo with Fixed Nodes Chang, Carlson,Pandharipande and Schmidt, PRL 91, 050401 (2003)

9. Fixed node GFMC results, S.-Y. Chang et al. PRA 70, 043602 (2004)

10. BCS →BEC crossover Leggett (1980), Nozieres and Schmitt-Rink (1985), Randeria et al. (1993),… If a<0 at T=0 a Fermi system is a BCS superfluid If |a|=∞ and nr031 a Fermi system is strongly coupled and its properties are universal. Carlson et al. PRL 91, 050401 (2003) If a>0 (ar0)and na31 the system is a dilute BEC of tightly bound dimers

11. Carlson’s talk at Pack Forest, WA, August, 2007

12. Fermi gas near unitarity has a very complex phase diagram (T=0) Bulgac, Forbes, Schwenk, PRL 97, 020402 (2007)

13. Very brief/skewed summary of DFT

14. Density Functional Theory (DFT) Hohenberg and Kohn, 1964 Local Density Approximation (LDA) Kohn and Sham, 1965 particle density only! The energy density is typically determined in ab initio calculations of infinite homogeneous matter. Kohn-Sham equations

15. Kohn-Sham theorem Injective map (one-to-one) Universal functional of density independent of external potential

16. How to construct and validate an ab initio EDF? • Given a many body Hamiltonian determine the properties of • the infinite homogeneous system as a function of density • Extract the energy density functional (EDF) • Add gradient corrections, if needed or known how (?) • Determine in an ab initio calculation the properties of a select number of wisely selected finite systems • Apply the energy density functional to inhomogeneous systems and compare with the ab initio calculation, and if lucky declare Victory!

17. One can construct however an EDF which depends both on particle density and kinetic energy density and use it in a extended Kohn-Sham approach (perturbative result) Notice that dependence on kinetic energy density and on the gradient of the particle density emerges because of finite range effects. Bhattacharyya and Furnstahl, Nucl. Phys. A 747, 268 (2005)

18. The single-particle spectrum of usual Kohn-Sham approach is unphysical, with the exception of the Fermi level. The single-particle spectrum of extended Kohn-Sham approach has physical meaning.

19. Extended Kohn-Sham equations Position dependent mass Normal Fermi systems only!

20. However, not everyone is normal!

21. Superconductivity and superfluidity in Fermi systems • Dilute atomic Fermi gases Tc 10-12–10-9eV • Liquid 3He Tc 10-7 eV • Metals, composite materials Tc10-3 – 10-2 eV • Nuclei, neutron stars Tc105 –106 eV • QCD color superconductivity Tc107 – 108 eV units (1 eV 104 K)

22. Bogoliubov-de Gennes equations and renormalization

23. SLDA - Extension of Kohn-Sham approach to superfluid Fermi systems Mean-field and pairing field are both local fields! (for sake of simplicity spin degrees of freedom are not shown) There is a little problem! The pairing field  diverges.

24. Why would one consider a local pairing field? • Because it makes sense physically! • The treatment is so much simpler! • Our intuition is so much better also. radiusofinteraction inter-particleseparation coherence length size of the Cooper pair

25. Nature of the problem at small separations It is easier to show how this singularity appears in infinite homogeneous matter.

26. Pseudo-potential approach (appropriate for very slow particles, very transparent, but somewhat difficult to improve) Lenz (1927), Fermi (1931), Blatt and Weiskopf (1952) Lee, Huang and Yang (1957)

27. The SLDA (renormalized) equations Position and momentum dependent running coupling constant Observables are (obviously) independent of cut-off energy (when chosen properly).

28. Superfluid Local Density Approximation (SLDA) for a unitary Fermi gas

29. The naïve SLDA energy density functional suggested by dimensional arguments

30. The renormalized SLDA energy density functional

31. How to determine the dimensionless parameters a, b and g ?

32. One thus obtains:

33. Bonus! Quasiparticle spectrum in homogeneous matter solid/dotted blue line - SLDA, homogeneous GFMC due to Carlson et al red circles - GFMC due to Carlson and Reddy dashed blue line - SLDA, homogeneous MC due to Juillet black dashed-dotted line – meanfield at unitarity Two more universal parameter characterizing the unitary Fermi gas and its excitation spectrum: effective mass, meanfield potential

34. Extra Bonus! The normal state has been also determined in GFMC SLDA functional predicts

35. Fermions at unitarity in a harmonic trap within SLDA • and comparison with ab intio results • GFMC - Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007) • FN-DMC - von Stecher, Greene and Blume, arXiv:0705.0671 • arXiv:0708.2734

36. Fermions at unitarity in a harmonic trap GFMC - Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007) FN-DMC - von Stecher, Greene and Blume, arXiv:0705.0671

37. NB Particle projection neither required nor needed in SLDA!!!

38. SLDA - Extension of Kohn-Sham approach to superfluid Fermi systems universal functional (independent of external potential)

40. Densities for N=8 (solid), N=14 (dashed) and N=20 (dot-dashed) GFMC (red), SLDA (blue)

41. New extended FN-DMC results D. Blume, J. von Stecher, and C.H. Greene, arXiv:0708.2734

42. New extended FN-DMC results D. Blume, J. von Stecher, and C.H. Greene, arXiv:0708.2734

43. Agreement between GFMC/FN-DMC and SLDA extremely good, • a few percent (at most) accuracy • Why not better? • A better agreement would have really signaled big troubles! • Energy density functional is not unique, • in spite of the strong restrictions imposed by unitarity • Self-interaction correction neglected • smallest systems affected the most • Absence of polarization effects • spherical symmetry imposed, odd systems mostly affected • Spin number densities not included • extension from SLDA to SLSD(A) needed • ab initio results for asymmetric system needed • Gradient corrections not included

44. Outlook Extension away from unitarity - trivial Extension to (some) excited states - easy Extension to time dependent problems – appears easy Extension to finite temperatures - easy, but one more parameter is needed, the pairing gap dependence as a function of T Extension to asymmetric systems straightforward (at unitarity quite a bit is already know about the equation of state)